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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION OF PENDULUMS.                         87
error of less than the fortieth part of a second in the observation of an interval of time amounting to 4-J- hours. If the apparent defect, amounting to about 0*04 or 0'05, in the theoretical result be real, it may be attributed with probability to an error in the correction for the wire. This would be no objection to the theory, for it will be remembered that the theory itself indicated the probable failure of the formulae generally applicable to a long ' cylinder when the cylinder comes to be of such extreme fineness as the wires employed in pendulum experiments.
58. The preceding experiments of Baily's are the most important for the purposes of the present paper, inasmuch as they were performed on pendulums of simple and very different forms ; but there still remain three sets of experiments, the fourteenth, fifteenth, and sixteenth, in which the pendulum consisted of a combination of a sphere and a rod, so that the results can be compared with theory. The details of these experiments being suppressed, I have been obliged to calculate the time of oscillation from the ordinary formulae of dynamics, but the results will no doubt be accurate enough for the purpose required. In all the calculations I have supposed the rod to reach up to the axis of suspension, and have consequently added 1*55 inch (the length of the shank of the knife-edge apparatus) to the length of the rod, and have added to the weight of the rod a quantity bearing to the whole weight the ratio of 1*55 inch to the whole length.
In the case of the spheres attached to the ends of the rods (sets 14 and 16) the process of calculation is as follows. Let I be the length of the rod increased by 1'55 inch, W1 its weight, increased as above explained, a the radius and W the weight of the sphere, A, the length of the isochronous simple pendulum. Then supposing the masses of the rod and sphere to be respectively distributed along the axis, and collected at the centre, which will be quite accurate enough for the present purpose, and putting a for the ratio of a to I, we have by the ordinary formula
iFI + (l+a)»TF
*•-                                           ............... (   ''
whence r, the time of vibration, is known.    The formula (148) then gives k, which applies to the sphere, and (147) gives m, the a